From $\mathcal{N}=2$ supersymmetry in four dimensions to (0,2) supersymmetry in two dimensions

We consider $\mathcal{N}=2$ four dimensional field theories compactified on a two torus in the presence of a $U(1)$ magnetic field. We discuss the restrictions leading to theories with (2,2) supersymmetry or (0,2) supersymmetry in two dimensions. The field theories live on D5 branes wrapped on four cycles of Calabi-Yau 3-folds or 4-folds described as resolved ADE singularities or resolved conifold fibered over a two torus.